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This is a short non-technical review focusing on the $\mathcal{W}_N$ family of $\mathcal{W}$-algebras and on their relation to quantum integrability. It is a summary of recently given seminars and workshop contributions.

We study the higher spin algebras of two-dimensional conformal field theory from the perspective of quantum integrability. Starting from Maulik-Okounkov instanton R-matrix and applying the procedure of algebraic Bethe ansatz, we obtain infinite commuting families of Hamiltonians of quantum ILW hierarchy parametrized by the shape of the auxiliary torus. We calculate explicitly the first five of these Hamiltonians. Then, we numerically verify that their joint spectrum can be parametrized by solutions of Litvinov's Bethe ansatz equations and we conjecture a general formula for the joint spectrum of all ILW Hamiltonians, based on results of Feigin, Jimbo, Miwa and Mukhin. There are two interesting degeneration limits, the infinitely thick and the infinitely thin auxiliary torus. In one of these limits, the ILW hierarchy degenerates to Yangian or Benjamin-Ono hierarchy and the Bethe equations can be easily solved. The other limit is singular but we can nevertheless extract local Hamiltonians corresponding to quantum KdV or KP hierarchy. Litvinov's Bethe equations in this local limit provide an alternative to Bethe ansatz equations of Bazhanov, Lukyanov and Zamolodchikov, but are more transparent, work at any rank and are manifestly symmetric under triality symmetry of $\mathcal{W}_{1+\infty}$. Finally, we illustrate analytic properties of the solutions of Bethe equations in minimal models, in particular for Lee-Yang CFT. The analytic structure of Bethe roots is very rich as it captures the representation theory of $\mathcal{W}_N$ minimal models.

We investigate the algebra of vector fields on the sphere. First, we find that linear deformations of this algebra are obstructed under reasonable conditions. In particular, we show that $hs[\lambda]$, the one-parameter deformation of the algebra of area-preserving vector fields, does not extend to the entire algebra. Next, we study some non-central extensions through the embedding of $\mathfrak{v}\mathfrak{e}\mathfrak{c}\mathfrak{t}(S^2)$ into $\mathfrak{v}\mathfrak{e}\mathfrak{c}\mathfrak{t}(\mathbb{C}^*)$. For the latter, we discuss a three parameter family of non-central extensions which contains the symmetry algebra of asymptotically flat and asymptotically Friedmann spacetimes at future null infinity, admitting a simple free field realization., Comment: 18+5 pages, 1 figure. Attached Mathematica notebook. v3: discussion added on page 11 and reference added

We study the 3-parametric family of vertex operator algebras based on the unitary Grassmannian coset CFT $\mathfrak{u}(M+N)_k/(\mathfrak{u}(M)_k \times \mathfrak{u}(N)_k)$. This VOA serves as a basic building block for a large class of cosets and generalizes the $\mathcal{W}_\infty$ algebra. We analyze representations and their characters in detail and find surprisingly simple character formulas for the representations in the generic parameter regime that admit an elegant combinatorial formulation. We also discuss truncations of the algebra and give a conjectural formula for the complete set of truncation curves. We develop a theory of gluing for these algebras in order to build more complicated coset and non-coset algebras. We demonstrate the power of this technology with some examples and show in particular that the $\mathcal{N}=2$ supersymmetric Grassmannian can be obtained by gluing three bosonic Grassmannian algebras in a loop. We finally speculate about the tantalizing possibility that this algebra is a specialization of an even larger 4-parametric family of algebras exhibiting pentality symmetry. Specialization of this conjectural family should include both the unitary Grassmannian family as well as the Lagrangian Grassmannian family of VOAs which interpolates between the unitary and the orthosymplectic cosets., Comment: ancillary Mathematica notebook for character calculations included

We study the even spin $\mathcal{W}_\infty$ which is a universal $\mathcal{W}$-algebra for orthosymplectic series of $\mathcal{W}$-algebras. We use the results of Fateev and Lukyanov to embed the algebra into $\mathcal{W}_{1+\infty}$. Choosing the generators to be quadratic in those of $\mathcal{W}_{1+\infty}$, we find that the algebra has quadratic operator product expansions. Truncations of the universal algebra include principal Drinfe\v{l}d-Sokolov reductions of $BCD$ series of simple Lie algebras, orthogonal and symplectic cosets as well as orthosymplectic $Y$-algebras of Gaiotto and Rap\v{c}\'{a}k. Based on explicit calculations we conjecture a complete list of co-dimension $1$ truncations of the algebra.

We construct a quadratic basis of generators of matrix-extended $\mathcal{W}_{1+\infty}$ using a generalization of the Miura transformation. This makes it possible to conjecture a closed-form formula for the operator product expansions defining the algebra. We study truncations of the algebra. An explicit calculation at low levels shows that these are parametrized in a way consistent with the gluing description of the algebra. It is perhaps surprising that in spite of the fact that the algebras are rather complicated and non-linear, the structure of their truncations follows very simple gluing rules.

We study the relation between $\mathcal{W}_{1+\infty}$ algebra and Arbesfeld-Schiffmann-Tsymbaliuk Yangian using the Maulik-Okounkov R-matrix. The central object linking these two pictures is the Miura transformation. Using the results of Nazarov and Sklyanin we find an explicit formula for the mixed R-matrix acting on two Fock spaces associated to two different asymptotic directions of the affine Yangian. Using the free field representation we propose an explicit identification of Arbesfeld-Schiffmann-Tsymbaliuk generators with the generators of Maulik-Okounkov Yangian. In the last part we use the Miura transformation to give a conformal field theoretic construction of conserved quantities and ladder operators in the quantum mechanical rational and trigonometric Calogero-Sutherland models on which a vector representation of the Yangian acts., Comment: no figures

We study the structure of modules of corner vertex operator algebras arrising at junctions of interfaces in $\mathcal{N}=4$ SYM. In most of the paper, we concentrate on truncations of $\mathcal{W}_{1+\infty}$ associated to the simplest trivalent junction. First, we generalize the Miura transformation for $\mathcal{W}_{N_1}$ to a general truncation $Y_{N_1,N_2,N_3}$. Secondly, we propose a simple parametrization of their generic modules, generalizing the Yangian generating function of highest weight charges. Parameters of the generating function can be identified with exponents of vertex operators in the free field realization and parameters associated to Gukov-Witten defects in the gauge theory picture. Finally, we discuss some aspect of degenerate modules. In the last section, we sketch how to glue generic modules to produce modules of more complicated algebras. Many properties of vertex operator algebras and their modules have a simple gauge theoretical interpretation.

We associate vertex operator algebras to $(p,q)$-webs of interfaces in the topologically twisted $\mathcal{N}=4$ super Yang-Mills theory. Y-algebras associated to trivalent junctions are identified with truncations of $\mathcal{W}_{1+\infty}$ algebra. Starting with Y-algebras as atomic elements, we describe gluing of Y-algebras analogous to that of the topological vertex. At the level of characters, the construction matches the one of counting D0-D2-D4 bound states in toric Calabi-Yau threefolds. For some configurations of interfaces, we propose a BRST construction of the algebras and check in examples that both constructions agree. We define generalizations of $\mathcal{W}_{1+\infty}$ algebra and identify a large class of glued algebras with their truncations. The gluing construction sheds new light on the structure of vertex operator algebras conventionally constructed by BRST reductions or coset constructions and provides us with a way to construct new algebras. Many well-known vertex operator algebras, such as $U(N)_k$ affine Lie algebra, $\mathcal{N}=2$ superconformal algebra, $\mathcal{N}=2$ super-$\mathcal{W}_\infty$, Bershadsky-Polyakov $\mathcal{W}_3^{(2)}$, cosets and Drinfeld-Sokolov reductions of unitary groups can be obtained as special cases of this construction.

We revisit the construction of multi-centered solutions in three-dimensional anti-de Sitter gravity in the light of the recently discovered connection between particle worldlines and classical Virasoro conformal blocks. We focus on multi-centered solutions which represent the backreaction of point masses moving on helical geodesics in global AdS$_3$, and argue that their construction reduces to a problem in Liouville theory on the disk with Zamolodchikov-Zamolodchikov boundary condition. In order to construct the solution one needs to solve a certain monodromy problem which we argue is solved by a vacuum classical conformal block on the sphere in a particular channel. In this way we construct multi-centered gravity solutions by using conformal blocks special functions. We show that our solutions represent left-right asymmetric configurations of operator insertions in the dual CFT. We also provide a check of our arguments in an example and comment on other types of solutions., Comment: 29 + 5 pages. v2: error in right-moving stress tensor corrected, section 4 expanded, references added

We discuss the representation theory of non-linear chiral algebra $\mathcal{W}_{1+\infty}$ of Gaberdiel and Gopakumar and its connection to Yangian of $\hat{\mathfrak{u}(1)}$ whose presentation was given by Tsymbaliuk. The characters of completely degenerate representations of $\mathcal{W}_{1+\infty}$ are for generic values of parameters given by the topological vertex. The Yangian picture provides an infinite number of commuting charges which can be explicitly diagonalized in $\mathcal{W}_{1+\infty}$ highest weight representations. Many properties that are difficult to study in $\mathcal{W}_{1+\infty}$ picture turn out to have a simple combinatorial interpretation., Comment: JHEP version

We study the operator product expansions in the chiral algebra $\mathcal{W}_{\infty}$, first using the associativity conditions in the basis of primary generating fields and second using a different basis coming from the free field representation in which the OPE takes a simpler quadratic form. The results in the quadratic basis can be compactly written using certain bilocal combinations of the generating fields and we conjecture a closed-form formula for the complete OPE in this basis. Next we show that the commutation relations as well as correlation functions can be easily computed using properties of these bilocal fields. In the last part of this paper we verify the consistency with results derived previously by studying minimal models of $\mathcal{W}_{\infty}$ and comparing them to known reductions of $\mathcal{W}_{\infty}$ to $\mathcal{W}_N$. The results we obtain illustrate nicely the role of triality symmetry in the representation theory of $\mathcal{W}_{\infty}$., Comment: 60 pages, 2 figures

We construct a class of smooth solutions in three-dimensional Vasiliev higher spin theories based on the gauge algebra hs[\lambda]. These solutions naturally generalize the previously constructed conical defect solutions in higher spin theories with sl(N) gauge algebra, to which they reduce when \lambda is taken to be equal to N. We provide evidence for their identification with specific primary states of the W_\infty [\lambda] algebra in a particular classical limit. In terms of the Gaberdiel-Gopakumar-'t Hooft limit of the W_N minimal models, this limit corresponds to a regime where the 't Hooft coupling becomes large., Comment: 15 pages + appendices. V2: minor corrections and clarifications, references added

We propose a refinement of the Gaberdiel-Gopakumar duality conjecture between W_N conformal field theories and 2+1-dimensional higher spin gravity. We make an identification of generic representations of the W_N CFT in the semiclassical limit with bulk configurations. By studying the spectrum of the semiclassical limit of the W_N theories and mapping to solutions of Euclidean Vasiliev gravity at \lambda=-N, we propose that the `light states' of the W_N minimal models in the 't Hooft limit map not to the conical defects of the Vasiliev theory, but rather to bound states of perturbative scalar fields with these defects. Evidence for this identification comes from comparing charges and from holographic relations between CFT null states and bulk symmetries. We also make progress in understanding the coupling of scalar matter to sl(N) gauge fields., Comment: 39 pages + appendices. v2: references added, minor modifications

We construct charged anisotropic AdS domain walls as solutions of a consistent truncation of type IIB string theory. These are a one-parameter family of solutions that flow to an AdS fixed point in the IR, exhibiting emergent conformal invariance and quantum criticality. They represent the zero-temperature limit of the holographic superfluids at finite superfluid velocity constructed in arXiv:1010.5777. We show that these domain walls exist only for velocities less than a critical value, agreeing in detail with a conjecture made there. We also comment about the IR limits of flows with velocities higher than this critical value, and point out an intriguing similarity between the phase diagrams of holographic superfluid flows and those of ordinary superconductors with imbalanced chemical potential., Comment: 11 pages, 3 figures. V2: Very minor corrections. JHEP version

We construct fully backreacted holographic superfluid flow solutions in a five-dimensional theory that arises as a consistent truncation of low energy type IIB string theory. We construct a black hole with scalar and vector hair in this theory, and study the phase diagram. As expected, the superfluid phase ceases to exist for high enough superfluid velocity, but we show that the phase transition between normal and superfluid phases is always second order. We also analyze the zero temperature limit of these solutions. Interestingly, we find evidence that the emergent IR conformal symmetry of the zero-temperature domain wall is broken at high enough velocity., Comment: v3: Published version. Figures 5 and 6 corrected. 24 pages, 7 figures

We study direct currents in a simple holographic realization of a superconducting film. We investigate how the presence of a DC current affects the superconducting phase transition, which becomes first order for any non-vanishing value of the current, as well as several other properties of the superconductor such as the AC conductivity. Near the critical temperature we find a quantitative agreement with several properties of Ginzburg-Landau superconducting films, for example the squared ratio of the maximal and minimal condensate is equal to two thirds. We also comment on the extension of our construction to holographic Josephson junctions., Comment: 22 pages, 7 figures; v2: typos corrected, ref and several clarifications added, figure 4 improved, published version; v3: typos corrected